In Refs. [1]}, [2]}, the original formulation of SCET has been extended to an effective theory called SCET\(_{\rm BSM}\) , which describes light SM fields coupled to a field describing a hypothetical new heavy resonance with mass much above the electroweak scale. In these papers the operator basis was constructed up to third (and partially fourth) order in the expansion parameter \(\lambda \)  for the case of a scalar resonance \(S\)  transforming as a singlet under the SM gauge group. In the present work we extend the framework to the case of heavy vector resonances, while at the same time allowing for the resonance to carry SM charges. An additional subtlety in this case arises from the fact that theories with massive vector fields are a priori not renormalizable. The mass of the resonance must either be acquired via a Higgs-like mechanism, or the resonance must arise as a composite state from a confining strong interaction. Neither of these mechanisms are visible to the effective theory. Hence, the vector resonance needs to be described in a fashion analogous to the \(b\)  quark in Heavy-Quark Effective Theory (HQET) [3]}, [4]}, [5]}, in which only the soft fluctuations of the heavy field remain as dynamical degrees of freedom. In this way, the effective Lagrangian is renormalized even though the vector resonance can appear as a (soft) virtual particle in loop diagrams. For most of this work we focus on the case of a vector resonance transforming as a SM singlet, which is commonly referred to as a \(Z^{\prime }\)  boson. In this case no loop-amplitudes with virtual heavy vector resonances exist. Even then, however, introducing an effective field for the heavy vector meson is needed to restrict the operator basis and keep the EFT power counting consistent, because different components of the vector field obey different scaling rules in the EFT.
